Maybe find an application of the subject that they might find interesting. I suppose if you can't find anything that interests them, then it's much harder to teach it.
For instance, perspective drawing might provide a nice application of 3D projective space, its subspaces, and perspectivities between those subspaces. Some of the theory of conic sections might be relevant too.
Computer graphics provides a nice application of coordinate geometry. This covers elementary algebra, Pythagoras's theorem, etc.
Even eating pizza can provide an application of differential geometry.
I tell my kids they can have letter cookies if they pick a word that starts with the letter, and can have 5 treats if they ask for 4 but know what "plus one means" or can have 4 if they recite "2 plus 2 equals ... ".
They're 3, so I don't expect that to scale, but I'm hoping it's normal reward-for-knowledge by the time we get report cards.
When I was having trouble learning multiplication my father made up a payment system. He made flash cards and I got a payment for every one I mastered (I had to get it right some number of times, not just once). I ended up with maybe $25 or $50 which was a lot for a kid in the 1970s.
My mother tried to give me $5 for every book of the Bible I read. I never took her up on it even though I knew about the basically freebies like Jude. I wasn't opposed to it, but it felt like –on the one hand– I didn't want to half-ass it and read a few books –and on the other– I really didn't want to read the entire Bible. So I guess that a completionist attitude prevented me from getting $30!
Kids do not understand the concept of "importance". At least no kid I've met. That part of their brain doesn't work. They'll trade effort for privileges or toys tho, and are little mimicry machines so they follow you if you use it.
yeah, I recoiled when the author of the post says "no bribing" - bribery is one of the most useful tools a parent has. I guess you could call it "incentive" or something, but really, it's quid pro quo.
Honestly it's so close to how the world works I can't believe 1. Avoiding loss of privilege and 2. Gaining new things as reward isn't the top two parenting tips.
But probably zeroth, most important, is modelling good behavior. Kids are mirrors.
We are torn but decided to keep books scarce, and scold them a little bit when they sneak books into bed. We know but they are much more likely to get their rebellion out reading now. Which I love.
We went the other way and have plentiful books, constant library trips etc. but have had to regulate sleep by setting a regular lights out time. Executive function and mood really suffers when sleep is insufficient.
I appreciate the idea of harnessing rebellion and I'll think more on how to apply that to my parenting :)
It's not about forcing your kids to "do math", but to excel at important skills far before the benefits of being good at that skill matter.
The amount of homework/study per day that maximizes math scores on tests is significant, 1+ hours/school day by the time they're in middle school, with it helping even more for those who are starting out poor at math[0]. You'll note the referenced study doesn't even max out progress for any group - meaning most could have studied more and improved more.
I don't know any kids that voluntarily did an hour or more of solely math study per day. I know plenty that were forced, and ended up loving math or other technical fields as adults.
As a parent of young kids, obviously I haven't gone through high school yet - but I don't think many children who reach their potential in math, english, music etc will have no pressure from their parents.
2. A love of subversives and cheating the system. Basically, the guys writing leetcode cheating software are saints in my book. All subversions of the attempt to turn society into a meritocracy (a term which was originally supposed to be a slur/negative connotation - https://en.m.wikipedia.org/wiki/The_Rise_of_the_Meritocracy) is extremely good.
3. An advanced knowledge of TI basic, so I could cheat hard on every single school assignment I could get away with. AP Chemistry? I’ve got a symbolic stoichometry solver app! Calculus? CAS system in the palm of my hand!
Play stupid games with children, win stupid prizes. Maybe don’t force them to work like little slaves in their early life, and they won’t strike back at your society systems.
Raising kids is hard. Sad. And what do I know about it? Regardless, parents do need to get involved in their children's education. For instance, they should help their children prepare for entry exams into secondary school. This shouldn't take their child 4 hours a day. Maybe 10 minutes on some days, and 1 hour on others.
I had similar conclusions, but the other way around: absolutely no guidance. Fortunately by the time I programmed and sold the basic ti math exam solvers to by classmates for 2 euros a pop I had everything memorized.
Nothing like cheating the system to know the system
Your experience sounds awful but surely there is a reasonable middle ground between forcing a kid to do any math and forcing them to do it in 4+ hour sessions.
You realize your experience can't be generalized to anybody else except for those who were abused in the same way you were? It also isn't what people in these comments are suggesting should be done.
> As a parent of young kids, obviously I haven't gone through high school yet - but I don't think many children who reach their potential in math, english, music etc will have no pressure from their parents.
Well it depends. I had no pressure from my parents to learn about programming but still got really good at it. Could I have gotten even better had I been pressured to "practice"? Perhaps. But then I also wouldn't like it for the reasons I do (I like making stuff, but not solving riddles) and it would feel like the dad sport situation.
I also played the piano for 6 years, starting out because I liked it. My parents didn't suggest it, but a few years in they were pushing me to continue even when I didn't like it anymore. Finished the first level of music school (6 years where I live) and haven't touched it since. Just to clarify, they weren't using any directly abusive tactics to keep me going, but they did put a lot of pressure onto it.
There's a lot of nuance to all of this and I don't completely disagree that we should occasionally pressure our kids to push their limits. What we often fail to acknowledge is that kids easily change their minds after a while. Just because they liked something at a certain point doesn't mean they still do. The easiest way to get a kid to dislike something is to make it a chore. Additionally, I think we need to ask ourselves whether it's more important to us to have a kid that's average scoring but has a (mostly) stress free upbringing, or one that excels but is stressed out by the time they hit high school. Kids absorb stress differently than we (adults) do.
I'm fortunate enough that my daughter has an admirable interest (and talent) for Math since very early age. She even won a medal at a renowned nationwide Math competition when she was in Grade 5... competing in the Grade 10 category.
The only thing better than telling a lie and people believing it, is telling the truth and no one believing it. Shows that her achievements are truly remarkable
My daughter and loads of kids watched number blocks from around two or three up, I think it made quite a big difference- she's far ahead of where I was now, years later.
ChatGPT makes it so easy to build a lesson/workbook for something your kid is interested in. I've used it to build workbooks on special relativity, tsolkovsky's rocket equation (including euler integration to build a scratch program), triangulation, logic gates, probabilities of simple dice games, etc. My pro-tip is to tell the LLM to format the document in LaTeX, so you get beautiful math typesetting.
You don't even have to get through the workbook. Get to a part that they need to understand better and make a detailed workbook on that part (for example, triangulation -> solving a system of linear equations).
I guess these workbooks usually come in two different "shapes" - one, a guided workbook with a high-level goal that combines several concepts, and another would be a practice worksheet where we do a bunch of exercise of the same algorithm (say, long division, calculating summations, or matrix multiplication) over and over. For the "workbook" pattern, we first discuss with the LLM the final goal (e.g. a scratch program that can calculate a rocket's position using the rocket equation). Then we flesh out the steps towards the goal - is it reasonable to add the math for air resistance? air resistance that decreases with altitude? gravity turn? how do we integrate the velocity and position for each frame? How can we relate the integration by step size to the underlying integral, by showing that the result gets more precise (but slower to calculate) the smaller your delta-t is? Then, produce a Scratch code sample that implements the velocity and position calculation. Of course, there are things subtly wrong with the code sample (usually, if the math formulas are well-known, they're correct), which requires debugging - just another type of problem-solving.
The second shape, worksheets, is a lot more straightforward. Just define the type of problem you want to practice and have chatgpt make a bunch of problems. Then switch to one of the newer reasoning models and have it work the problems, and refine to get rid of any bogey problems (for example, for polynomial exercise, you could tell it to make sure the roots are integers)
The worksheets are more "hands off" - I run them through the algorithm once and check their work once or twice and then let them do the rest. The important thing is that the worksheets are connected to their high-level goal, and they understand that in order to solve the big, hairy problem that they're interested in, they need to build up certain specific skills.
Usually the worksheet goal is a pretty substantial conceptual stretch for my kids so they need to go through a series of fundamental worksheets. But the great thing about the LLM is, you can just tell it you're having a problem understanding some concept and to help build the scaffolding by listing all of the required skills to understand a concept, and picking the ones that needs improvement the most and practicing them.
My approach draws a little from "The MathAcademy Way" - https://www.justinmath.com/files/the-math-academy-way.pdf but instead of building fundamentals evenly in all topics before advancing (like expanding a sphere), we look only at the scaffolding required to support some higher-level goal - it's sort of like the masters/PhD process but guided through existing human knowledge: https://www.openculture.com/2017/06/the-illustrated-guide-to... . As a side note, I think it's really fun to include the history (mathematicians who contributed to the ideas) as well as the notation (using the greek letters, explaining why it's common to use them). When the kids notice the names like Pythagoras, Newton, and Euler reappearing frequently, and get a sense of the time scale these discoveries happened on, they treat the current state - and their ability to go learn thousands of years of math in months - with more reverence.
Another article where someone thinks being a parent means they understand all children. I have 4 kids, and 2 of them definitely would never do math, even basic math, for fun, ever. Their brains lack whatever pathways most people utilize to learn math, so I now have a 15 year old who has to work nearly as hard at arithmetic as she did when she first learned it. No amount of drilling, change of curriculum, buckling down or backing off has had any impact. She has absolutely no interest in math. But the kid reads faster than I do, which is not slow at all.
The only thing I know about kids after having adopted 4 of them is that none of them are alike. The only time when you can really train them to do anything consistently is when they are babies. As a result all 4 have great sleep habits :)
Yes. And it's the same when the kids come from the same parents too. We have one kid that's willing to go very deep on math. The only does what can be figured out in 3 seconds or less. Same genetic parents, same school system.
The original concept in the article of exploration is great. Some kids want to explore math, some science, some music, and some Starcraft.
I read it (as a non-child), and a lot of my certainties about what young brains are and are not capable of got joyfully exploded. I'm not linking it to you proscriptively, or with a specific suggestion or riposte in mind whatsoever - you just might be interested in it.
School is very successful at convincing kids (and former kids) of two things: firstly, that the academic subjects they purport to teach are actually delineated by the school textbooks and curricula. And secondly, that the reaction people have to specific subjects within these school structures are the actual unchangeable nature of the person's relationship with the subject.
I hope one day our societies move past these two egregious and immeasurably damaging beliefs.
I haven’t read the book, but we have 100% had the “you don’t have to graduate if you don’t want to” talk with this one haha. She doesn’t want to drop out, but definitely isn’t interested in college. We want to keep that door open for her if we can, so we just remind her that staying in school requires doing some things she doesn’t like.
But yeah, at 15 it gets a little hairy. You have a kid who wants to be an adult, but in a lot of ways they are not prepared to make adult decisions still. Eventually she will have to make them, ready or not. But we have a few years left to help her, so the focus becomes how to best do that.
> School is very successful at convincing kids (and former kids) of two things:
Well, and thirdly, that your worth as a person is determined by your results in graded examinations, and by extension, your salary or some other numerical rating decided by someone else.
I as a child, a teen, and a young adult thought I hated math, I got bad grades and it bored me. I dropped out of school. I later went to college and took remedial algebra twice.
Math in school was purposeless and rigid, a rote procedure to be followed by command because that's what kids have to do.
Now, I have grown older, and my curiosity drove me to learn because I wanted to make things, machines and software and probabilistic strategies. Things that necessitate math. If you can't rotate a vector, your guy walks faster diagonally. If you can't think mathematically and you want to lift a 2 jointed robot arm that weighs several tons, you're going to tip it over, and possibly die in the process. You can do it without trig but you can't do it without thinking about math.
Once I found purpose, I began to appreciate the beauty of the more elegant solutions. I kind of fell in love with math as an adult. Now I watch numberphile with my kids and make complicated machinery and software at work.
I think a lot more people love math than realize it, because they're conflating math itself and what school calls math, which is worksheets and demands, not beauty and creation.
With my kid in elementary school, I can see how math instruction is generally terrible: teachers rarely have any enthusiasm for teaching it. I only had one great math teacher (combining enthusiasm, skill and hard work) and I've been through special math programs (in high school and uni).
Again, it is a question of incentives: someone with enthusiasm for math would likely go with a higher paying job requiring higher level math.
Still, despite the crappy teachers, I was better than most to persevere at it until high school where I had the great teacher.
But this does not scale and we are losing kids to bad teachers: how can we fix this?
I don’t personally see how one person’s experience with children other than my own has any connection to my own children. That was the point I was attempting to make, though. Just because you have anecdotes to share doesn’t mean you’ve stumbled upon some universal truth. They can be helpful to share but NOT if used to dismiss other people’s experience.
I knew some who were bad at math. Asian immigrant test scores on math are ~1/2-1 standard deviation higher than white Americans. That’s noticeable comparing groups of people but still leaves a lot of Asian immigrants who are not good at math.
There is no royal road. If all your kids are biologically yours and you and all your family are good at math and you marry someone from a similar family, you can stack the deck maybe 95/5 in favor or your kid being good at math? But that option is already off the table if you lack that talent. And there are other things you should probably prioritize first!
The key element here is nurturing curiosity.
Since then i and my 10yr old have been sitting through a virtual math circle led by Aylean McDonald on parallel.org.uk an organization run by Simon Singh
For elementary school age kids, maybe even middle school, try getting them started with the app "Euclidea".
They won't think of it as math. It's gamified geometric constructions. Starts simple, "how do you bisect an angle" with a compass and a straight edge. It goes to a very high level that will challenge anyone.
Something we don't pay enough attention to is that while calculators have solved everyday math to the point we downplay it as a required skill, people are not pulling out their calculator at the grocery store to make better purchase decisions, even though we all have one in our pocket now.
So we handwave the importance of being able to do everyday math in our heads, while also not taking advantage of the tool that's a substitute for it. We're less educated but also less effective than we would be if we'd never invented automated calculation and were forced to be sharp about it.
Is there a name for this phenomenon?
And what's it going to look like a decade after AI has caused people to stop using their brain for general thinking like it's stopped them from doing math?
(I'm sure you, the reader, are very good at math and are an exception to this still-apt generalization.)
But the reality is that's usually almost a false trade - I'm not buying one item in the grocery store I'm buying easily a dozen or more. The best way to do this would be toss the online inventory into a solver to calculate "best value" for me, but in reality it would be a waste of time because if they're out of something, or the quality looks suspect, then that blows that calculation completely. And then am I going to do this for every single item, where every minute in the store is multiplying through the rest of my day? How much is the time shopping trading off against extremely sparse leisure time?
And then there's intangibles - something being slightly cheaper doesn't necessarily mean I'm making a good trade off by buying it for my overall quality of life.
In Australia at least this whole problem was perfectly adequately solved by mandating bulk price labeling on all items in the supermarket. Products in comparable categories have per volume/weight prices listed alongside item prices.
> In Australia at least this whole problem was perfectly adequately solved by mandating bulk price labeling on all items in the supermarket. Products in comparable categories have per volume/weight prices listed alongside item prices.
Same in Europe. Mandatory labeling per volume/weight, pre-discount, after-VAT in addition to the discounted item prices. Then you glance at the shelves and make up your mind.
Learning to get to a best price per unit is a pretty useful skill that could make a lot of difference for a lot of future adults, just like financial literacy.
Some things aren't optional, and if they are seen as such, it's going to force the child to learn later in life what they couldn't earlier on.
I think most people don't care about optimizing an extra $10 out of their weekly grocery run.
Probably a better example is figuring out the cost of a loan. Just multiply the amortized monthly payment by the term and compare that to the loan amount. If the difference makes you balk, then go ahead and walk.
How many people even realize that loan interest is a significant cost and would bother to do that? Or know how to do that? Most people just try to minimize monthly payments to something they can bear and sign the paperwork.
> I think most people don't care about optimizing an extra $10 out of their weekly grocery run
I remember when this kind of "optimization" was done regularly by a great many shoppers on budgets. Back in the day some stores even used to put calculators on the shopping carts.
People used to know how to budget. Apparently the average American is affluent enough to not need to be able to do this any more. I worry that the atrophy of these kinds of practical skills will cause much pain for a great many people at some point down the road.
I worry too. American consumer credit card debt sounds like it's rising pretty quickly, and many people I talk to have a lack of free will and carry a sense of unfairness in the system they live in.
However, people can also adapt pretty quickly.
Those grocery shopping "optimization" skills are making a big come back (and have been since Covid). There are plenty of YouTube and TikTok videos popularizing how to get more out of their grocery hauls.
Lots of people are also learning how to budget, how to invest, etc. and sharing their excitement about it too. For some folks, they finally learn this stuff in their 40s and 50s, but there are also a lot of young adults learning these skills thanks to the Internet.
So I also have hope.
I just don't think the lack of basic math and budgeting skills displayed by average consumers are a problem so much as a symptom.
30 minutes of play per 3 days is such a brutal reality to acknowledge. One of the most wonderful experiences in all of life so drastically limited by the society we’ve constructed.
In some ways yes, but men have always been the ones to go hunt/farm for long hours and provide for the family, leaving the children home under the care of the mother/village for days or weeks at a time.
I would go so far as to say modern society actually enables us to be more involved in our children’s lives, especially those for whom remote work and home schooling are options.
To clarify, for those who seem not to have RTFA but are downvoting (I can only assume based on perceived sexism since nobody has been brave enough to comment)—-the parent was quoting an article written by a working father, who said his internal KPI was, “If I haven’t spent 30 minutes playing with my kids in the last 3 days something is wrong.”
> One of the most wonderful experiences in all of life so drastically limited by the society we’ve constructed.
I could understand if someone was forced to work two full-time jobs (as my grandfather was), but I find it much harder to blame ‘society’ when so many of these situations are self-imposed.
It’s possible that I’m jaded from hearing a subset of parents complain about not having enough time with their kids but then get stuck scrolling their phone while kids want to play. I also know some parents who insist on having a spotlessly clean house every day and then complain that there is enough time to spend with their kids.
I’ve gravitated toward peer parents who have similar priorities in life which has indirectly made me happier. Seeing all of the parents in my friend circles prioritize spending time with their kids and being honest with themselves about their priorities has been unexpectedly helpful for my own sanity.
Again, nothing against parents who are really forced to allocate time elsewhere, but I’m tired of seeing self-inflicted problems of prioritization and time management be externalized as blaming society.
Compare learning math to learning to bicycle. There is some some sweat and struggle that needs to be put in, before one "gets it". After this it can become enjoyable. I encouraged my daughter with practice exercises from a young age, but tried to avoid making it a drudge. She built up confidence and did well with it. She is also very hands on creative. She decided to study engineering and is working towards her PhD.
Those aren't nearly comparable. Riding a bike is one simple skill and as long as you're not racing that's enough for most people. Meanwhile learning maths is a years-long effort at best. I learned how to ride a bike within an hour by myself when I finally had a good reason to learn it. I can't say the same about maths.
Bike is fabulous self-correcting vehicle in most operation conditions. The trick really is just to learn to trust it when it is moving. And then what to do when it stops.
Math is layers upon layers upon layers. And then it also branches. Never really had willpower to learn it myself alone.
It does depend on what one defines as "riding a bicycle." At a moderate pace from point A to point B, or simply a lap around the neighborhood is one thing. Technical single tracks, grueling climbs, steep and slippery downhills, racing, maintenance and tuning, trials, jumps, drafting...
Math is just splitting the bill with your friends, seeing if you have enough cash for a hotdog AND a coke, or counting how many months of savings in order to buy that 5000$ bicycle.
20" competition trials bike, mulleted DH racing machine, full-squish slope-style dirtjumps, Japanese keirin, multi-week bikepacking, ITT, freeride.... Obviously there is some overlap.
Math, for most people, is the same a bicycles, for most people. With a handful of simple concepts, you can get by daily life.
I tell my kid that math is a language. You learn to speak it, just like you learn to speak any other language, slowly, by listening, understanding, speaking, intuitively recognizing patterns, rules and exceptions. When you start to become fluent you translate problems into math and solve them. At school they keep trying to make them memorize useful phrases, like a tourist that goes to Paris and learns how to say "where's the bathroom", "hello", "would you like to sleep with me", "thank you", "goodbye", etc.
"At school they keep trying to make them memorize useful phrases, like a tourist that goes to Paris…."
Like learning dozens of trig identities without any explanation about why one would need them. As I've mentioned elsewhere learning math for the sake of it isn't enough. For most of us math has to have relevance, and for that we have to link it to things in the real world.
If you are a physicist or an economist, you may be using mathematics as a language in the sense that you are using a mathematical description to convey an understanding of the natural world or the economy to your colleagues. But if you are a mathematician, you are interested in the mathematical objects for their own sake.
There is also a difference between the purpose of learning language and learning math. The goal of learning language is (often) to be fluent in it. In other words, the goal is to reach a level of proficiency which would allow you to not have to think about language and focus on the content of the conversation instead. On the other hand, the goal of learning mathematics is usually to be able to solve mathematical problems. Being able to do math without "thinking about it" is not usually a requirement.
Forcing is kind of hopeless. So is logic, and reasoning.
How children learn (rely on the prefrontal cortex of their adults) is very different than how adults learn (no fully developed prefrontal cortex until 25-26), learning about this can help a lot.
Learning more about the Reggio Emelia approach might help parents curious about this, it has been quite surprising how much is possible naturally. One of the best things to do is to relentlessly read to and with your kids.
Also, linking a topic to their interest's radar, encouraging curiosity, play in general, and letting them potentially discover it can go a long away.
When they've got something they want, teaching math and savings is a great thing. Understanding life is a lot harder without knowing a basic bit of math, and can be made a bit easier when doing it younger.
I had a math teacher that once made it clear, some stuff can just click, others is just about doing a lot of examples to learn the patterns. Doing math is very different than being creative with being comfortable to find it.
Today, I'd probably setup a good prompt to find a way for the child to share their mine to discover how they like to learn, and how they might like to learn faster and easier by taking some shortcuts through math directly or on navigating an ontology/taxonomy perspective.
Forcing is kind of hopeless. So is logic, and reasoning.
How children learn (they can't rely on a fully formed prefrontal cortex like adults) is very different than how adults learn (no fully developed prefrontal cortex until 25-26), learning about this can help a lot.
Learning more about the Reggio Emelia approach might help parents curious about this, it has been quite surprising how much is possible naturally. One of the best things to do is to relentlessly read to and with your kids.
Showing kids the math in every day things, especially things they already love is a helpful way of making it approachable, or at least aware.
Also, linking a topic to their interest's radar, encouraging curiosity, play in general, and letting them potentially discover it can go a long away.
When they've got something they want, teaching math and savings is a great thing. Understanding life is a lot harder without knowing a basic bit of math, and can be made a bit easier when doing it younger.
I had a math teacher that once made it clear, some stuff can just click, others is just about doing a lot of examples to learn the patterns. Doing math is very different than being creative with being comfortable to find it.
Today, I'd probably setup a good prompt to find a way for the child to share their mine to discover how they like to learn, and how they might like to learn faster and easier by taking some shortcuts through math directly or on navigating an ontology/taxonomy perspective.
When I was 8, I went to the library in our town a lot. My parents went there sometime to return their books. At some point I just stayed there when they would go home. First I was in the children/teenager section and soon in the general library, where I would read about programming and computers. I learned C by age of nine.
The "undeveloped PFC" argument is shallow, unspecific and usually just used to infantilize younger people. It may be useful if the child is under 6 years old, but at the time someone is 17 or older, it becomes essentially useless.
My learning process was always, and still is fueled by curiosity.
> One of the best things to do is to relentlessly read to and with your kids.
My mom would bring us into the clubhouse in the backyard and read to us, which I found really boring. Ended up not liking books because of it, and I'm pretty sure the same happened to both of my brothers. For years I'd only read the bare minimum required for school.
Years later I happened to see an neat book cover in the impulse-buy section of a store and begged for the book. That one book was what actually got me set on reading, and from then on I'd always have something with me.
She never realized this and still thinks I like reading because she read to us. I can't help but wonder how many of the anecdotes here are also parents not realizing what's actually going through their kids' minds.
I would agree. Kumon wasn't my son's favorite thing and he eventually decided/asked to stop. But the repetition and discipline of working at it every day had an influence. It didn't manifest until undergraduate but he ended up switching majors to math and is now pursuing a math PhD. Probably not common, but like learning to play the piano, I think it gave him a comfort with the basics and an intuition that allowed him to explore his own interests.
As someone who just finished school, I’m trying to figure out how to get genuinely interested in mathematics. I’ve never been particularly strong at it, yet I’m planning to enter a university program that demands a high level of math. The problem is, it’s hard to motivate myself to study math for its own sake. For example, I loved learning programming because it’s hands‑on—I can build something and immediately see the results. In everyday life, though, I rarely need more than basic arithmetic or simple sin/cos/tan trigonometry.
How do you develop a lasting interest in math when it doesn’t feel immediately useful?
I'm going to share my anecdote, because it may help, but everyone is different and your mileage may vary.
I'm a MechE by classical training (professionally I actually work doing software/network stuff, don't ask, DNS (screams internally)), so here's where it stood out for me:
Internalize what this simple example represents, think about why that's mathematically interesting, and start looking for where it applies elsewhere. You too could be roped into doing systems engineering at scales you didn't think people haven't already figured out.
One of my undergrad degrees is in math. As you study it, you learn to identify your assumptions (axioms), find or build interesting abstractions, prove properties about them (theorems), and then map all sorts of other things into those abstractions by figuring out that they're really the same thing. It's even more interesting when you start to find things that are different or question things you always took for granted.
Math gives you the ability to leverage the very structure and relationships of pure abstraction. It's quite the super power.
None of the specific things you learn studying math will be nearly as useful as the ability to think mathematically.
N=1 datapoint here.
I studied physics in university and before I started I was not aware that physics is basically just math where the results sometimes relate to reality. The pure math courses I took were the most difficult and in the beginning I loathed them, because it felt so unattainable to get any intuition, let alone real proper comprehension for all the concepts they threw at us. For a long time I felt like I was just hanging on by threads and especially if I compared myself to those who had some innate interest in math or generally some really good intuition on the abstract concepts (or even prior knowledge) it was really demotivating. But I also felt like I had no choice but to continue and as time went on the I grew fond of it. And the feeling of being overwhelmed changed - that is to say I still was completely lost every time a new topic was breached and I could not understand even half of the proofs in class - but I did not feel so defeated about it. And I grew to like the feeling of actually completing the work sheets they gave us every week. The process of solving them was often excruciating but if you did the sense of accomplishment is real. I think for most people higher math is really difficult and that is part of why it is interesting. Another aspect I had to accept over time is that even though you can state a mathematical fact or conjecture in just a hand full of symbols or a plain sentence it does not mean that truly understand it, its implications or how you got there can be understood the same way that other prose can be. Sometimes you have to stare at, contemplate and scribble around one equation for days until you understand whats up.
If there was any advice I would give, then it's probably similar advice on how to stop procrastinating on anything that is difficult. Establish a routine first - find a spot that you will only use for studying this (like a spot in a library), start small, divide and conquer, accept that you will not understand most things easily, reward yourself for the small wins along the way, find an accountability partner or someone to study with if that's your thing, make a regular schedule with regular times where this is what you do - consistency is key, even if its just for 5 minutes, stack it onto other habits, see yourself as a scholar of math - it is what you do, lean into the discomfort, as enduring that is a valuable skill in itself.
Perhaps think of them as solving logical puzzles. It's fun. Even though not always related to everyday tasks.
For me, it began many years ago when reading about Hilbert's hotel paradox. Turns out our laymen's understanding about infinity isn't as really refined.
I write mobile apps for living and indeed these stuffs are irrelevant for my work.
Probability/Statistics is a good excuse to learn mathematics, because paying a little attention one finds lots of day-to-day situations where is possible to apply it. For example, see the secretary problem[1].
I wish I couldve excluded everything past basic algebra and hopped right to statistics at a young age - I *loved* everything about the practicality of it, how it explained tangible relationships and illuminated the world. Algebra and calculus were so un-engaging I had those teachers calling me everything but a stupid child.
Have you ever watched a video of a highly skilled tetris player? Where they fill the screen most of the way to the top and then suddenly they just combo the whole thing down and everything wraps up cleanly, and then they start fresh.
The feeling of "oh yeah, that was nice watching that mess turn into something clean and squared away" is where I get a lot of my joy from math.
But also, there are uses to math that you might be able to play with through every day, but you've never thought of those scenarios in a mathematical way.
I was walking today, and on the street there is a right angle turn. The inner portion of the turn is just a square right angle, but the outside of the turn is a radius. I started wondering to myself, if I want to be on the outside of the turn going into and exiting the turn, what would be different ways I could walk this, and what would the distance differences be.
Crossing directly across, to the inner corner and crossing directly across to the outer side again, would be 2w (for the width of the road w). Following the edge of the radius would (assuming perfectly circular), be 1/4 of a circle, so 1/42piw = 1/2 pi * w. The shortest route is a straight line, which would make a right triangle, so w^2 + w^2 = c^2, 2w^2 = c^2, sqrt(2) w = c
So crossing twice is 2w, following the edge is 1/2piw, and shortest path is sqrt(2)*w. Not super applicable, and I didn't need to do math to figure it out, but I was walking and bored, so I found joy in it. The fact that they all boil down to having w as a factor means I could figure out a nice ratio between all of them. And then I needed to mentally figure out what 1/2 pi was. 3.14/2 = 1.57... And I know that sqrt(2) is roughly 1.41 ish.
So now I know that crossing twice has a cost of 2, following the edge is 1.57, and direct line is 1.41. Following the edge is vaguely close enough to the ideal path to warrant not walking into the street to optimize the route, 1.57 / 1.41 is about ~110%. Whereas by defintion, a cost of 2 is going to be sqrt(2) times sqrt(2), so ~141% more than shortest path.
A few things to note here. First off, I'm aware that not everyone finds the same joy in doing simple mental math and thinking about problems mathematically even when there is no need to do it, but trying to think of things more minor trivial things mathematically may cause you to at least appreciate it more, which can grow into joy. And second, I wasn't doing any complicated math in my head. I just thought to myself "is it faster to cut to the inside corner and then cut back out... of course not, right?" and I was able to answer that definitively to myself. Did it matter? Was the answer probably obvious anyway? Probably, but I was able to _prove_ that. And I value facts. Finding joy in the simple things lets you build up more of a familiarity and view it more as a problem solving tool than a tedious thing to rote memorize.
A great way to build up math familiarity and see how other people find joy in mathematics would be to watch Numberphile videos on YouTube[0]. It's a bunch of mathematicians sharing things they find interesting about math. Some times are REAL hard to grasp, but some are just very interesting puzzles[1]. The puzzles don't always have clear immediate usefulness, but can often be described as "a mathematician wanted to know an answer, so they did some math to find out and prove something to themself."
Sorry, end of spiel.
tl;dr - find the joy in the simple things and use math as a tool to answer (even simple) questions to help highlight the usefulness.
I didn't particularly find (at the time) calculus, multivariable calculus, physics, etc. interesting as I didn't find the applications interesting at the time. I find these subjects representative of what you traditionally learn at school.
When I entered uni I discovered my passion for discrete math, algebra (groups, rings, fields, etc.), number theory, cryptography, theory of computation, etc. as they have a lot of application in CS.
That's really what did it for me - and also I had great uni lecturers. I wish they would have taught the subjects I like in highschool - the difficulty level is about the same.
Make it practical! Graphics programming involves linear algebra. Databases involve relational algebra. Machine learning involves requires calculus. You’ll naturally encounter hands-on tasks with tangible goals that involve learning new math.
as someone who loved maths first but then do programming for a living, it's because solving puzzles is fun. I get the same dopamine hit whether it's a math problem, a coding task, or a video game level. but I think forcing yourself to like something is not the correct approach; you either like it naturally or you tolerate it for some other goals
University is not a good place for learning mathematics as most of your math instructors there will be very good at math and very bad at teaching people that are not already very good at math.
Except maybe not calculus. I remember my calc class kind of being terrible because it was a weed out for other majors. Every math class that wasnt required though was great.
For whatever reason, many University programs use high level math classes as a filter to weed out 1st year students from that program. If university instructors had a genuine passion, and ability, for teaching high level math then they wouldn't accept that as an outcome.
That is unfortunately true; not only in the US, but all around the world. The particulars do depend on the instructor, and many if not most instructors try to be motivational, but the syllabus is perfectly clear: "this is a weed out class". And when it comes to test time, the syllabus wins.
The only thing I disagree with in your comment is about the instructors: they want to be employed, and they have to accept the syllabus and testing standards. It is not about passion and ability to teach (most, especially younger ones, are full of those); it is about meeting the departmental requirements.
You are misrepresenting what's happening. Other departments use beginning math classes as a way of weeding out students they feel won't succeed in their fields because they can't pass basic mathematics classes. Most math departments would absolutely love to have more students in them.
The problem is that these students aren't prepared properly by K-12 mathematics courses and math builds upon itself. If you don't have a good grasp of algebra, you just won't succeed at calculus. We're sticking people in the equivalent of Spanish 4 without having learned Spanish 1 properly.
> For example, I loved learning programming because it’s hands‑on—I can build something and immediately see the results. In everyday life, though, I rarely need more than basic arithmetic or simple sin/cos/tan trigonometry.
Consider doing something that actually needs it. You like computer programming - consider making a game engine. It might be easier to learn when you can actually see that it is useful.
Keep in mind though that math is a lot of things. People obsess over calculus but that is just one type. Math is just as much the different types of symmetry in wall paper patterns as it is finding the derrivative. Don't be afraid to try different areas. If you dont know where to start, consider picking up "A Concise Introduction to Pure Mathematics" by liebeck which introduces a bunch of different math concepts and see if any feel more interesting to you.
If you love programming, there's quite a lot of programming where math is vital. Graphics, optimisation problems, cryptography, neural networks, figuring out if a hash works, projecting if an algorithm will scale...
The tricky bit is often that you need to learn some of the math before you can see how it's useful, but if you need stronger motivation, you might try diving into a slightly math heavy programming problem and learn the math as you go
Don't study it for usefulness, study it for beauty. Look for amazing insights.
Yes, you need some practical math as well. I did engineering, there's a lot of inelegant stuff there.
But that stuff actually tends to be right next to some very interesting things.
Here are three things you can find out.
First, there's more than one kind of infinity. You can't make a map from natural numbers like 1, 2, 3 etc to real numbers like e, 0.632268, sqrt(2) etc. Look for Cantor diagonalization.
Second, a random walk like a heads vs tails comes back to zero almost certainly. It also does so in two dimensions, like walking randomly in Manhattan. In three dimensions, it does not, and so for higher dimensions. Look for Polya.
Third. There is a way for you and me to communicate secretly, despite everyone in HN being able to see our entire exchange. Look for Diffie Helmann.
These days, there's a whole industry of people doing math videos with interesting stuff.
It's easier to appreciate math when you are disinterested in the results or applications, because the nature of academic topics near the core grouping of math/philosophy/empiricism is that they are discovered with a lot of meandering at first, and then sometime down the line they become repurposed into a direct application that can be learned by rote. School tends to instruct in some of the most directly applicable stuff first - the "three R"s" plus some civics and training aligned with national goals. And that means that school predominantly teaches associations between math and rote methods, to the disgruntlement of many mathematicians. The "meandering" part is left to self-selected professionals, so it doesn't get explored to much depth.
So I think a good motive for math study is really in games and puzzles, where the questions posed aren't about win/lose or right/wrong, but about exploring the scenario further and clarifying the constraints or finding an interesting new framing. Martin Gardner wrote a long-running column and a few books in this vein which are still highly regarded decades later.
I feel like a lot of platitudes are being said here.
If anyone had a guaranteed way to make people enjoy math, we'd already be applying that method.
Just read ahead to figure out what you'll need to learn, and do some advance reading. Anything thag make the courses easier will tend to make them more fun.
A friend of mine taught remedial math at UW to incoming freshmen. She would write:
x + 2 = 5
on the blackboard and ask a student "what is the value of x?" The student would see the x, and immediately respond with x means algebra, algebra is hard, I cannot do algebra.
So she started writing:
_ + 2 = 5
and ask the student to fill in the blank. "Oh, it's 3!"
I’ve always found that an indictment of math education — and spent many, many hours discussing it.
When teaching addition, workbooks commonly use a box, eg, “[ ] + 2 = 5” — and first graders have no conceptual problem with this. Somehow, we lose people by the time we’re trying to formalize the same concept in algebra. There’s been many times I’ve written a box around letters in a problem and asked students “what’s in the box labeled x?”
I agree. I think the actual problem is that the student is trying to comprehend what it means for anything to have mathematical value other than explicit numbers.
Numbers and letters are taught together, but not as symbols. Letters are taught with sounds and numbers are taught with counting. The notion of a symbol isn't really emphasized much.
I would explain it more like after
[ ] + 2 = 5
what happens if you need more than one box for a complicated problem? Teach the idea that saying box #3 is equivalent to assigning an arbitrary letter for whatever reason you want, but that people more familiar with math prefer letters because they stand in for words that describe what the number is for. You might want to use 'c' for the number of cats you're trying to figure out.
In a room of five animals two are dogs. How many cats?
a = 5, c = ?, d = 2
a = c + d
so... 5 = c + 2
what is c?
Light bulb goes off: "You can do that?" Yes, you can do whatever you want and it's not all about carrying the one or whatever other rote teaching they've been given. They can get creative and be engaged, and then you let them know that actually there are some conventions people like to use for what they're trying to do. They might even believe they've invented a new idea. At least they're having fun.
To me, a lot of pre-college math education could be summarized as "In this class I will show you a bunch of abstract problems, a bunch of ways to solve them, and I will test if you have learned them." Learning in these classes is often limited to memorizing a sequence of steps.
That's why I would frequently ask "You can do that?" myself when talking to those whom I considered mathematically gifted (math olympiad winners and such). I think they realized that as a problem-solving tool math could be used creatively. I saw it as a largely useless hammer that to work had to be held in a very specific way.
I remember connecting sets in, I think, Pascal to what I had learned in school and realizing that all that math was perhaps not as useless as I had thought : - )
You skipped a step. One of the problems is more obvious with a different operator we learn when in the "box" stage:
> Go from "[ ] x 2 = 10" to writing it "box x 2 = 10--what is box?". Then "b x 2 = 10--what is b?" then "x x 2 = 10--what is x?".
From memory, we didn't switch from "x" to dot for multiplication until at the exact same time we started using symbols. If we'd done it earlier (or even right from the start) it might not have been as much of a problem.
most likely this very unfortunate misnomer started with fortran, where it was deemed lucrative to point out "how much programs look like mathematic formulas!".
not only is this overloading a symbol (equality) with a completely different meaning (assignment), it is also a poor choice typographically, as it represents a directional operation with a directionless symbol.
using an arrow for assignment is much better.
it's also worth pointing out that unlike most others, logic programming languages (e.g. prolog) have actual variables, not references to mutable or immutable memory cells.
arrow for assignment is cool, but the backspace key is the only closest arrow-like key on the keyboard but it has a different purpose. plus the arrow key should be laid out such that u dont have to press a SHIFT/CTRL/ALT to produce it.
for this reason, i felt C a breath of fresh air cos u could just assign using = instead of what we was doing in pascal which was the horrible := where u had to press SHIFT for the :
The semantic meaning of a blank is much better understood to everyone than an arbitrary letter like 'x'.
People just want to know why it's x and not something else or how a letter can have value. They might even think how can 24 + 2 = 5? They just want something to grab onto and nobody is really teaching the concept of a symbol in a math class.
The problem is, in many MANY MANY schools, teachers are more like social workers that have to compensate for utter horrifics outside of school. You got a ton of children so poor they didn't have breakfast which means their first (and all too often: only) meal will be the school-provided lunch (Covid showed that - a bunch of schools were open at least for lunches). You got children that are literally homeless and living with their parents in some car on a Walmart parking lot. You got children whose parents are in and out of jail. You got children living with their siblings in way too small, pest and mold ridden "apartments". You got children whose parents don't have money to pay for basic school supplies. You got children who are dealing with mental, physical and sexual abuse. You got children where the parents are constantly on drugs or seeking for drugs. You got children with a drug dependency on their own - if they're lucky it's just tobacco or weed, if not it's opioids. You got children with parents or siblings with serious mental or physical health issues. Or you got children with their own mental and physical health issues, or if you want it worse, children with these issues but without access to any kind of treatment. You got children that are being weaponized in nasty divorces. You got children that are being weaponized by street gangs. You got children committing crimes from petty theft to dealing drugs just to survive. You got children that have to literally work (and states like FL pushing to have more working children). You got children having their own children already (either from sexual abuse, from under-education about their own bodies, or intentionally because they fell for some stupid challenge/dare). You got children dealing with bullying, you got some who actually are bullies because they have no other way of dealing with their emotions or getting lunch money. You got children with parents with about zero interest in them. You got children who worry that they'll come home and find out their parents got snatched and disappeared by ICE. You got children who worry that ICE will storm their classroom and deport them. You got children who worry they might not survive the school day because someone will shoot at them. You got children who are constantly on the move because their parents' employment/deployment requires absolute mobility. You got children who are LGBT and have to deal with ever increasing hate against them (and LGBT youth already had significantly higher suicide rates than before the GQP made it a culture war issue).
The US doesn't have any kind of system to help these children but schools and libraries, both are horribly underfunded (there's some school districts where teachers gotta take up second jobs because the government can only afford paying them for 4 days a week), and all too often teachers have to pay with their own money for students' school supplies.
And on top of dealing with these kind of nightmares, they actually have to try and teach these children something - even if the children in question aren't anywhere near a headspace where they can actually learn.
I taught at a nice middle-class school, so most of the problems you mention were not relevant there. And yet, it seemed like half of the kids' families were either recently divorced or in the process of divorce. I couldn't really blame those kids for not paying attention to school. And this seemed like the best case, so probably at most places it gets much worse.
Education has a problem with scaling, especially at the elementary level. Sometimes people figure out a nice solution, but when you tell them "great, and now do this in every village" the problem becomes obvious. But there are kids in that village, too, and you want them to know reading and math and hopefully also something more.
> Education has a problem with scaling, especially at the elementary level.
Not if you actually provide the money. Europe gets this down decently well - although I'll admit, in rural areas in Germany we got some serious consolidation issues thanks to urban flight.
But at least our teachers are well paid government jobs and the job is decently attractive.
There's plenty of money thrown at schools in the US, but the issue is that the students that live in poor socioeconomic conditions tend to not do well. The "simple answer" that addresses the root cause would make individuals not subject to poverty and whatnot. But throwing money at institutions is already on the ropes in the US, let alone throwing money at the "undeserving".
(Yes, this is a political opinion. No, do not blame me for that. Politics does not come wrapped up neatly with a bow tie in a box. If you want to debate the veracity of my claim, go do that instead.)
> If you want to debate the veracity of my claim, go do that instead.
I'd do no such thing because you are completely correct - the only thing I'd add is that poverty, while being very dominant, isn't the only issue that desperately needs to be fixed.
Scaling is not just about money. It's also about teachers. You can have hundreds of great teachers, but if the entire educational system requires tens of thousands of teachers, you will end up with many mediocre ones, because you simply don't have tens of thousands of great ones.
This becomes even more true in higher level maths where programming language style functions would make everything vastly more clear, and easily typeable, than the traditional Greek symbols. sum(x+3, 1, 4) is just so much more clear (and consistent when generalized across other operations) and practically as concise as the mathematical way of expressing that which I cannot even type. Multiple variables would be a bit dirtier, but still much cleaner than the formal expression.
Interestingly mathematical symbols in the past also regularly evolved. Then at some point we just stopped doing that and get stuck in a time which is arguably no longer especially appropriate. So we're left with rather inconsistent symbols, oft reused in different contexts, and optimized for written communication.
The formal language of math is intensely optimized for rapidly communicating with yourself 90 seconds in the future, when doing a proof or calculation, turning paper into working memory. It does seem silly to use the same language for communicating with others across unkniwn but deep chasms of context. Its remarkable that it works at all
its silly. itd be like introducing first year programming students to advanced maps/filters/anonymous function syntax, instead of the easier to understand for loops and if/else statements. math's "no true scottsman" approach to teaching only hurts itself in the long run.
I'm not sure if it would be easier to explain a map / filter to a first year student vs implementing the patten manually using a for loop and if statement...
Seems like a pretty easy example to make practically, for map have a collection of things, say balls or black. Pick up each one and do a thing to them, paint them blue for example.
For filter do the same except have two different colour balls, if they are yellow they get thrown away, of they are blue they get put in a bucket.
A for loop doing exactly the same you would need to explain the topic at hand, as well as explain iterating an index etc...
Explaining loops is independent of the concepts of collections though. It's also more general, since map/filter/reduce use some kind of loops under the hood anyway, the fact that probably shouldn't be ignored in education process. Unless of course you go with pure functional recursive iterator, but good luck explaining that one.
Maps and filters also require understanding of higher order functions and the very idea of passing function around as a value. I would argue that implementing map/filter with a loop and then demonstrating how this pattern is generalized as .map()/.filter() functions is better and more accessible
The strangest part about mathematics culture is that there is a culture of vibing the notation.
Nobody in school ever tells you that there are glossaries on Wikipedia that tell you the meaning of the symbols. You're supposed to figure it out yourself using vibes.
The way mathematic notation is taught is inherently unstructured. You're expected to just get it.
For the purposes of education, it is important to keep in mind that "optimized for performance of a highly trained person" and "optimized for understanding of a complete beginner" are two different things.
I often see people make the mistake of trying to teach inappropriately abstract things to small children, because that's what the pros do, and we want the little kids become pros as soon as possible. Problem is, trying to skip the fundamentals is only harmful in long term.
First kids need to learn what all that stuff means, and then we can proceed to teach them the shortcuts.
Thanks for this comment. My secret shame as a programmer is that I haven't really learnt much maths, stopped at 16 in school. Writing out the sum function like you did makes perfect sense to me immediately.
What I should really do is create myself a cheat sheet of symbols to code...
It's hard to debate that mathematical notation has a lot of room for improvement. High level algebra is very cryptic and often looks like an arcane incantation rather than something comprehensible for an unknowing person.
That said, as a person who moderately enjoyed math in high school and university, this functional notation would make me hate math infinitely more. It's would look like Lisp, which, at high level, looks just as cryptic as algebra. The sheer amount of braces and mistakes that would be made when reading and writing them is nauseating.
Infix notation, for all its flaws, provides important visual aid for understanding the structure of the expression (the sum of two fractions looks very different from fraction of two sums for example). Whereas with functional notation it's like working on linear textual representation of abstract syntax tree. Trust me, nobody wants to read, write or transform one by hand
I thought that was it too, but you're saying it's not..? I've been thinking about it for a few minutes now and I still can't figure out what other meaning it could have.
Edit: Oh wait, someone else mentions map/filter, did they mean this as a combination of range->map->sum and the latter two numbers are the range portion, like sum(map(x+3, 1..4)) ?
Edit2: And now I'm remembering sigma. I think it would have been more obvious to me if the order was flipped and your issue handled the way it is in that notation: sum(x=1, 4, x+3), though I'd still prefer the range notation: sum(x=1..4, x+3)
Yep. I agree and now we've basically reinvented sigma. Take the x=1 and put it below, take the 4 and put it above take the x+3 and put it to the right.
Granted I always found sigma a bit quirky for separating the range ends like that. Either x below and 1..4 above, or x=1..4 below/above would have been more intuitive.
But it's just a notation you learn once and then you know it.
I don’t recall the exact age, but when I was doing math in primary school (somewhere around age 9/10) we were absolutely using symbols - “Paul has two apples, and the basket can hold 10 apples. How many more apples can Paul put in the basket” is the same as 2 + x = 10
We did these sorts of problems for a long time, with addition/multiplication/fractions, and even when we started doing actual algebra the problems were introduced the same way “let’s look at a problem we’ve solved already, and write it in a different way”.
I will die on the hill that most of math would benefit from better naming, less short names and longer format.
Yes, the crack math guys have no problems with terse symbols. But most people do.
Good example is Greek letters for geometry. They are not really taught in school so an easy formula gets 'weird squiggly thing times another squiggly thing....' and that does not help understanding at all
For any given problem, you usually know what it is your studying, so writing out names doesn't have much benefit. On the other hand, a more visual language (which is what mathematical writing is) lets you easily look at specific portions of the picture and read off how it behaves, which is very useful. Basically, getting hung up on names means you're reading it wrong.
This is like trying to change English or arguing that we should all speak Esperanto. Mathematical notation isn't the way it is to save ink or make it look difficult. It's that way because it works. Notation isn't set by committee, it's just a way of communication that works. If you read cutting edge research you'll find notations being invented all over the place. Most of them will never go anywhere, some will become standard in their field (like big-O) and others will become universally used (like dropping the multiplication symbol and using epsilon for a small number).
I think this is a very limited take for a hacker forum. We talk about how useful accurate names for variables are all the time, or generally how working to encode more natural/context-related semantics to code helps anyone reading it understand what the goal is better than an extremely terse symbology.
Yeah, lots of existing math texts will forever exist with greek alphabet soup, but we don't have to rely on those as our be-all-end-all teaching tools.
To operate at a high level in mathematics I would agree that having the skill of easily abstracting complex things into compact symbols is a necessary skill, just as I would agree to the same concept applied to software engineering or really any complex engineering system; by the same token, we don't have to START on hard mode with all of our students. Math is infamously difficult for some, largely (I think) because we make it unnecessarily opaque out of some misguided sense of traditionalism.
If we want to have lots of people who are good at math we should embrace whatever pedagogy is effective.
Many programs are at a much higher level of abstraction than mathematics. If you are implementing domain logic then you should definitely use names from the domain. But when implementing an algorithm often the most meaningful name is a single character. I find it odd when people try to force the "no single character" rule everywhere.
But I've got to say, the short names are not the problem. If you rewrote F=ma as "force is equal to mass multiplied by acceleration" this wouldn't suddenly make it more accessible to swathes of the population. People who are good at maths anyway have no problem with this.
imagining that alpha is what stops people from being good at math is a useless take. why pretend that such a low bar could prevent anyone from understanding? its some fake generosity towards "newcomers" that is completely unwarranted. math notation is by and for professional mathematicians.
are you really saying that "let function(argument has type RealNumber) has type RealNumber be a function from a real number argument to a real number" is somehow superior to "let f(x) : R->R"
> why pretend that such a low bar could prevent anyone from understanding?
I don't know why this is a startling take. If you encode your ideas in an unfamilar symbology, of course that's going to make it more difficult for someone who isn't familiar with the space.
I'm not arguing that we should teach real analysis this way, or any other high level math class. I'm only contesting GP's comment that there is NO value to be had in using more familiar language to explain a new concept to an unfamiliar audience.
This is the entire reasoning behind "word problems" at the elementary level; they're meant to ground the abstract modeling of a math problem (193 - 3 * 12 = ?) into something more intuitive for a child to understand (If you start with 193 eggs, and you take three dozen away to bake a cake, how many are left?)
> are you really saying that "let function(argument has type RealNumber) has type RealNumber be a function from a real number argument to a real number" is somehow superior to "let f(x) : R->R"
No, I'm saying the there's tradeoffs on either side, and our educators ought to be aware of this.
> math notation is by and for professional mathematicians.
I agree, but we teach math to plenty of people who aren't professional mathematicians. I wouldn't want to do formal abstract algebra proofs in a more verbose form, I'm perfectly happy using the domain notation, but my friends from biological sciences who have to take a calculus course now have to learn both a new symbology alongside the problem domain. I've watched enough of my (clearly intelligent) biology friends slam face first into calculus and spin out. They're not dumb, they can do circles around me when it comes to chemistry, yet they Just Can't wrap their heads around calculus-style math, which leads me to wonder what the difference is between how we teach complex chemistry vs complex math. Questioning the pedagogy is a fairly logical extension of that.
If there was some better notation that allowed biology people to understand calculus more easily, what do you imagine they'd do with it?
It's hard for me to imagine it without any special notation. It makes me think of the general relativity in words of four characters or less that was posted recently. Sure, it might be possible, but does it really make it easier to understand? Understanding is normally built up in layers. We learn things using big words because that makes it easier than learning with small words. And we learn maths with funny symbols because it makes it easier than learning it with words (or colours or mime or other things you already know).
Math doesn't start on hard mode. Students spend years studying how numbers behave before doing symbolic math. That said, trying to cover the topics symbolic math does with a more verbose language would just make it into impossible mode. It'd be like trying to replace sheet music with words.
In fact the analogy to music notation is I think a fairly strong one. People's complaints always sound to me like asking why we don't write "C3 sixteenth note" for music instead of using dots and lines. After all, how are we meant to know what the dots mean and remember the difference between an eighth and sixteenth, or what flats/sharps do? And then the key signature can modify all of it!
The notation just isn't a barrier. Once you learn to read it, it's there because it's a clearer way to write the ideas. The hard part for people is they don't understand the ideas, and don't have the frameworks like key signatures, chord progressions, and meter to place them within. Longer words for variables won't help people understand e.g. inner and outer regular measures, or the open cover definition of compactness. That comes from a lot of work to understand what you're trying to say, the pitfalls of saying it wrong, and precisely how your slightly different way of saying it avoids those pitfalls (or selects the best set of pitfalls if you must pick some kind of degenerate behavior).
I hadn't thought of sheet music in this context before; that's a helpful counter-example, thanks.
Broadly I agree; the semantic density of domain-specific language is often required to operate well in that domain. I disagree some with the "Math doesn't start on hard mode," but I think that's just bikeshedding at some level.
The endemic "I just don't understand math" that my (American) peers have espoused, to me, points to a failure in our (American, public school) instruction practices around it.
> People just want to know why it's x and not something else or how a letter can have value.
The way I was taught it and the way that worked now for my now 3 year old is just to say pirates buried a number under the X, and that we need to guess what they buried. If the concept of a number being hidden is a barrier to understanding for anyone they have seriously bad teachers.
There is a game called dragonbox algebra which I'm currently working through with my son and is an absolutely fantastic approach to this problem. Sadly its now part of a horrendous subscription service and is hard to access. I find it really sad that we've had computers for decades and there are so few good maths games like this.
As a senior in high school, I devoured this game in elementary school and got way better at math than my peers. Now taking differential equations and multivariable calculus through our college in the high school (CHS) program. When I looked for it out of curiosity I was sad to see it transformed into a subscription service.
There are two sides to this. The system or method might be bad but also a determined person can go all the way and perform at a decent level if they put in enough time.
Even if the system was better the person still has to be able to motivate themselves and put in the time.
One of my school math teacher had the same approach in another way: We were expected to use greek letters, not latin ones.
Same reasoning: It showed us kiddos that the letter was insignificant compared to the concept expressed by the letter.
So my take would be: Your friend taught the students for the first time what they were actually doing while handling equations with "a letter in it". That is no problem of algebra in itself. It just means their previous teachers sucked.
She loves it. It uses a ‘?’ for basic algebra style problems and after a few days of playing (if/when she wants to, we don’t make her play it), she was already much better and faster at those problems. It made me think that schools should be giving kids games like these.
If you formulate the warnings just right, you wouldn’t need to “force” it, as kids will be willing to look both sides themselves. They like to be alive.
And then your kids and their same generation would be replaced by their peer kids from hard working boys and girls from India and China. Unfortunately curiosity only works with brilliant minds. Normal minds plus curiosity is useless.
I think the advice is good for younger children. The author is using 14+11 as an example. Very engaged parents can have a tendency to overdo it, so it's probably a good reminder.
As kids get older, they need to learn how to struggle and overcome struggles. (I would still caution against "forcing" math.) But yes, you need to start engaging hard work and determination.
Btw, the two are not mutually exclusive. Young children should be praised for struggling at things so they begin learning that skill, too.
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[ 2.5 ms ] story [ 340 ms ] threadFor instance, perspective drawing might provide a nice application of 3D projective space, its subspaces, and perspectivities between those subspaces. Some of the theory of conic sections might be relevant too.
Computer graphics provides a nice application of coordinate geometry. This covers elementary algebra, Pythagoras's theorem, etc.
Even eating pizza can provide an application of differential geometry.
They're 3, so I don't expect that to scale, but I'm hoping it's normal reward-for-knowledge by the time we get report cards.
But probably zeroth, most important, is modelling good behavior. Kids are mirrors.
My parents would reward me by letting me pick out a book at the book store. I'd be excited the whole week.
I may feel differently about reading it I had been forced to read and rewarded with something else, like junk food.
I appreciate the idea of harnessing rebellion and I'll think more on how to apply that to my parenting :)
The amount of homework/study per day that maximizes math scores on tests is significant, 1+ hours/school day by the time they're in middle school, with it helping even more for those who are starting out poor at math[0]. You'll note the referenced study doesn't even max out progress for any group - meaning most could have studied more and improved more.
I don't know any kids that voluntarily did an hour or more of solely math study per day. I know plenty that were forced, and ended up loving math or other technical fields as adults.
As a parent of young kids, obviously I haven't gone through high school yet - but I don't think many children who reach their potential in math, english, music etc will have no pressure from their parents.
[0]: https://pmc.ncbi.nlm.nih.gov/articles/PMC8025066/
1. A true hatred of work, make work, and a strong desire to defend laziness as a concept (note that Bertrand Russel agrees hard with me here!) -https://en.m.wikipedia.org/wiki/In_Praise_of_Idleness_and_Ot...
2. A love of subversives and cheating the system. Basically, the guys writing leetcode cheating software are saints in my book. All subversions of the attempt to turn society into a meritocracy (a term which was originally supposed to be a slur/negative connotation - https://en.m.wikipedia.org/wiki/The_Rise_of_the_Meritocracy) is extremely good.
3. An advanced knowledge of TI basic, so I could cheat hard on every single school assignment I could get away with. AP Chemistry? I’ve got a symbolic stoichometry solver app! Calculus? CAS system in the palm of my hand!
Play stupid games with children, win stupid prizes. Maybe don’t force them to work like little slaves in their early life, and they won’t strike back at your society systems.
Nothing like cheating the system to know the system
Well it depends. I had no pressure from my parents to learn about programming but still got really good at it. Could I have gotten even better had I been pressured to "practice"? Perhaps. But then I also wouldn't like it for the reasons I do (I like making stuff, but not solving riddles) and it would feel like the dad sport situation.
I also played the piano for 6 years, starting out because I liked it. My parents didn't suggest it, but a few years in they were pushing me to continue even when I didn't like it anymore. Finished the first level of music school (6 years where I live) and haven't touched it since. Just to clarify, they weren't using any directly abusive tactics to keep me going, but they did put a lot of pressure onto it.
There's a lot of nuance to all of this and I don't completely disagree that we should occasionally pressure our kids to push their limits. What we often fail to acknowledge is that kids easily change their minds after a while. Just because they liked something at a certain point doesn't mean they still do. The easiest way to get a kid to dislike something is to make it a chore. Additionally, I think we need to ask ourselves whether it's more important to us to have a kid that's average scoring but has a (mostly) stress free upbringing, or one that excels but is stressed out by the time they hit high school. Kids absorb stress differently than we (adults) do.
You don't even have to get through the workbook. Get to a part that they need to understand better and make a detailed workbook on that part (for example, triangulation -> solving a system of linear equations).
The second shape, worksheets, is a lot more straightforward. Just define the type of problem you want to practice and have chatgpt make a bunch of problems. Then switch to one of the newer reasoning models and have it work the problems, and refine to get rid of any bogey problems (for example, for polynomial exercise, you could tell it to make sure the roots are integers)
The worksheets are more "hands off" - I run them through the algorithm once and check their work once or twice and then let them do the rest. The important thing is that the worksheets are connected to their high-level goal, and they understand that in order to solve the big, hairy problem that they're interested in, they need to build up certain specific skills.
Usually the worksheet goal is a pretty substantial conceptual stretch for my kids so they need to go through a series of fundamental worksheets. But the great thing about the LLM is, you can just tell it you're having a problem understanding some concept and to help build the scaffolding by listing all of the required skills to understand a concept, and picking the ones that needs improvement the most and practicing them.
My approach draws a little from "The MathAcademy Way" - https://www.justinmath.com/files/the-math-academy-way.pdf but instead of building fundamentals evenly in all topics before advancing (like expanding a sphere), we look only at the scaffolding required to support some higher-level goal - it's sort of like the masters/PhD process but guided through existing human knowledge: https://www.openculture.com/2017/06/the-illustrated-guide-to... . As a side note, I think it's really fun to include the history (mathematicians who contributed to the ideas) as well as the notation (using the greek letters, explaining why it's common to use them). When the kids notice the names like Pythagoras, Newton, and Euler reappearing frequently, and get a sense of the time scale these discoveries happened on, they treat the current state - and their ability to go learn thousands of years of math in months - with more reverence.
The only thing I know about kids after having adopted 4 of them is that none of them are alike. The only time when you can really train them to do anything consistently is when they are babies. As a result all 4 have great sleep habits :)
The original concept in the article of exploration is great. Some kids want to explore math, some science, some music, and some Starcraft.
I read it (as a non-child), and a lot of my certainties about what young brains are and are not capable of got joyfully exploded. I'm not linking it to you proscriptively, or with a specific suggestion or riposte in mind whatsoever - you just might be interested in it.
School is very successful at convincing kids (and former kids) of two things: firstly, that the academic subjects they purport to teach are actually delineated by the school textbooks and curricula. And secondly, that the reaction people have to specific subjects within these school structures are the actual unchangeable nature of the person's relationship with the subject.
I hope one day our societies move past these two egregious and immeasurably damaging beliefs.
But yeah, at 15 it gets a little hairy. You have a kid who wants to be an adult, but in a lot of ways they are not prepared to make adult decisions still. Eventually she will have to make them, ready or not. But we have a few years left to help her, so the focus becomes how to best do that.
Well, and thirdly, that your worth as a person is determined by your results in graded examinations, and by extension, your salary or some other numerical rating decided by someone else.
Math in school was purposeless and rigid, a rote procedure to be followed by command because that's what kids have to do.
Now, I have grown older, and my curiosity drove me to learn because I wanted to make things, machines and software and probabilistic strategies. Things that necessitate math. If you can't rotate a vector, your guy walks faster diagonally. If you can't think mathematically and you want to lift a 2 jointed robot arm that weighs several tons, you're going to tip it over, and possibly die in the process. You can do it without trig but you can't do it without thinking about math.
Once I found purpose, I began to appreciate the beauty of the more elegant solutions. I kind of fell in love with math as an adult. Now I watch numberphile with my kids and make complicated machinery and software at work.
I think a lot more people love math than realize it, because they're conflating math itself and what school calls math, which is worksheets and demands, not beauty and creation.
Again, it is a question of incentives: someone with enthusiasm for math would likely go with a higher paying job requiring higher level math.
Still, despite the crappy teachers, I was better than most to persevere at it until high school where I had the great teacher.
But this does not scale and we are losing kids to bad teachers: how can we fix this?
There is no royal road. If all your kids are biologically yours and you and all your family are good at math and you marry someone from a similar family, you can stack the deck maybe 95/5 in favor or your kid being good at math? But that option is already off the table if you lack that talent. And there are other things you should probably prioritize first!
This is really stupid.
The key element here is nurturing curiosity. Since then i and my 10yr old have been sitting through a virtual math circle led by Aylean McDonald on parallel.org.uk an organization run by Simon Singh
They won't think of it as math. It's gamified geometric constructions. Starts simple, "how do you bisect an angle" with a compass and a straight edge. It goes to a very high level that will challenge anyone.
So we handwave the importance of being able to do everyday math in our heads, while also not taking advantage of the tool that's a substitute for it. We're less educated but also less effective than we would be if we'd never invented automated calculation and were forced to be sharp about it.
Is there a name for this phenomenon?
And what's it going to look like a decade after AI has caused people to stop using their brain for general thinking like it's stopped them from doing math?
(I'm sure you, the reader, are very good at math and are an exception to this still-apt generalization.)
And then there's intangibles - something being slightly cheaper doesn't necessarily mean I'm making a good trade off by buying it for my overall quality of life.
In Australia at least this whole problem was perfectly adequately solved by mandating bulk price labeling on all items in the supermarket. Products in comparable categories have per volume/weight prices listed alongside item prices.
Same in Europe. Mandatory labeling per volume/weight, pre-discount, after-VAT in addition to the discounted item prices. Then you glance at the shelves and make up your mind.
Some things aren't optional, and if they are seen as such, it's going to force the child to learn later in life what they couldn't earlier on.
Probably a better example is figuring out the cost of a loan. Just multiply the amortized monthly payment by the term and compare that to the loan amount. If the difference makes you balk, then go ahead and walk.
How many people even realize that loan interest is a significant cost and would bother to do that? Or know how to do that? Most people just try to minimize monthly payments to something they can bear and sign the paperwork.
I remember when this kind of "optimization" was done regularly by a great many shoppers on budgets. Back in the day some stores even used to put calculators on the shopping carts.
People used to know how to budget. Apparently the average American is affluent enough to not need to be able to do this any more. I worry that the atrophy of these kinds of practical skills will cause much pain for a great many people at some point down the road.
However, people can also adapt pretty quickly.
Those grocery shopping "optimization" skills are making a big come back (and have been since Covid). There are plenty of YouTube and TikTok videos popularizing how to get more out of their grocery hauls.
Lots of people are also learning how to budget, how to invest, etc. and sharing their excitement about it too. For some folks, they finally learn this stuff in their 40s and 50s, but there are also a lot of young adults learning these skills thanks to the Internet.
So I also have hope.
I just don't think the lack of basic math and budgeting skills displayed by average consumers are a problem so much as a symptom.
I would go so far as to say modern society actually enables us to be more involved in our children’s lives, especially those for whom remote work and home schooling are options.
I could understand if someone was forced to work two full-time jobs (as my grandfather was), but I find it much harder to blame ‘society’ when so many of these situations are self-imposed.
It’s possible that I’m jaded from hearing a subset of parents complain about not having enough time with their kids but then get stuck scrolling their phone while kids want to play. I also know some parents who insist on having a spotlessly clean house every day and then complain that there is enough time to spend with their kids.
I’ve gravitated toward peer parents who have similar priorities in life which has indirectly made me happier. Seeing all of the parents in my friend circles prioritize spending time with their kids and being honest with themselves about their priorities has been unexpectedly helpful for my own sanity.
Again, nothing against parents who are really forced to allocate time elsewhere, but I’m tired of seeing self-inflicted problems of prioritization and time management be externalized as blaming society.
Math is layers upon layers upon layers. And then it also branches. Never really had willpower to learn it myself alone.
Math is just splitting the bill with your friends, seeing if you have enough cash for a hotdog AND a coke, or counting how many months of savings in order to buy that 5000$ bicycle.
Math, for most people, is the same a bicycles, for most people. With a handful of simple concepts, you can get by daily life.
They will force themselves to play... and do math in the process.
Quite a story condensed into those five phrases.
Like learning dozens of trig identities without any explanation about why one would need them. As I've mentioned elsewhere learning math for the sake of it isn't enough. For most of us math has to have relevance, and for that we have to link it to things in the real world.
Instead we will just continue to slog along with the same poor system. Math wise, completely at the expense of the top down learners.
I think there are some differences
If you are a physicist or an economist, you may be using mathematics as a language in the sense that you are using a mathematical description to convey an understanding of the natural world or the economy to your colleagues. But if you are a mathematician, you are interested in the mathematical objects for their own sake.
There is also a difference between the purpose of learning language and learning math. The goal of learning language is (often) to be fluent in it. In other words, the goal is to reach a level of proficiency which would allow you to not have to think about language and focus on the content of the conversation instead. On the other hand, the goal of learning mathematics is usually to be able to solve mathematical problems. Being able to do math without "thinking about it" is not usually a requirement.
How children learn (rely on the prefrontal cortex of their adults) is very different than how adults learn (no fully developed prefrontal cortex until 25-26), learning about this can help a lot.
Learning more about the Reggio Emelia approach might help parents curious about this, it has been quite surprising how much is possible naturally. One of the best things to do is to relentlessly read to and with your kids.
Also, linking a topic to their interest's radar, encouraging curiosity, play in general, and letting them potentially discover it can go a long away.
When they've got something they want, teaching math and savings is a great thing. Understanding life is a lot harder without knowing a basic bit of math, and can be made a bit easier when doing it younger.
I had a math teacher that once made it clear, some stuff can just click, others is just about doing a lot of examples to learn the patterns. Doing math is very different than being creative with being comfortable to find it.
Today, I'd probably setup a good prompt to find a way for the child to share their mine to discover how they like to learn, and how they might like to learn faster and easier by taking some shortcuts through math directly or on navigating an ontology/taxonomy perspective.
How children learn (they can't rely on a fully formed prefrontal cortex like adults) is very different than how adults learn (no fully developed prefrontal cortex until 25-26), learning about this can help a lot.
Learning more about the Reggio Emelia approach might help parents curious about this, it has been quite surprising how much is possible naturally. One of the best things to do is to relentlessly read to and with your kids.
Showing kids the math in every day things, especially things they already love is a helpful way of making it approachable, or at least aware.
Also, linking a topic to their interest's radar, encouraging curiosity, play in general, and letting them potentially discover it can go a long away.
When they've got something they want, teaching math and savings is a great thing. Understanding life is a lot harder without knowing a basic bit of math, and can be made a bit easier when doing it younger.
I had a math teacher that once made it clear, some stuff can just click, others is just about doing a lot of examples to learn the patterns. Doing math is very different than being creative with being comfortable to find it.
Today, I'd probably setup a good prompt to find a way for the child to share their mine to discover how they like to learn, and how they might like to learn faster and easier by taking some shortcuts through math directly or on navigating an ontology/taxonomy perspective.
The "undeveloped PFC" argument is shallow, unspecific and usually just used to infantilize younger people. It may be useful if the child is under 6 years old, but at the time someone is 17 or older, it becomes essentially useless.
My learning process was always, and still is fueled by curiosity.
My mom would bring us into the clubhouse in the backyard and read to us, which I found really boring. Ended up not liking books because of it, and I'm pretty sure the same happened to both of my brothers. For years I'd only read the bare minimum required for school.
Years later I happened to see an neat book cover in the impulse-buy section of a store and begged for the book. That one book was what actually got me set on reading, and from then on I'd always have something with me.
She never realized this and still thinks I like reading because she read to us. I can't help but wonder how many of the anecdotes here are also parents not realizing what's actually going through their kids' minds.
How do you develop a lasting interest in math when it doesn’t feel immediately useful?
I'm a MechE by classical training (professionally I actually work doing software/network stuff, don't ask, DNS (screams internally)), so here's where it stood out for me:
https://en.wikipedia.org/wiki/Hydraulic_analogy
Internalize what this simple example represents, think about why that's mathematically interesting, and start looking for where it applies elsewhere. You too could be roped into doing systems engineering at scales you didn't think people haven't already figured out.
Math gives you the ability to leverage the very structure and relationships of pure abstraction. It's quite the super power.
None of the specific things you learn studying math will be nearly as useful as the ability to think mathematically.
If there was any advice I would give, then it's probably similar advice on how to stop procrastinating on anything that is difficult. Establish a routine first - find a spot that you will only use for studying this (like a spot in a library), start small, divide and conquer, accept that you will not understand most things easily, reward yourself for the small wins along the way, find an accountability partner or someone to study with if that's your thing, make a regular schedule with regular times where this is what you do - consistency is key, even if its just for 5 minutes, stack it onto other habits, see yourself as a scholar of math - it is what you do, lean into the discomfort, as enduring that is a valuable skill in itself.
For me, it began many years ago when reading about Hilbert's hotel paradox. Turns out our laymen's understanding about infinity isn't as really refined.
I write mobile apps for living and indeed these stuffs are irrelevant for my work.
[1] https://en.m.wikipedia.org/wiki/Secretary_problem
The feeling of "oh yeah, that was nice watching that mess turn into something clean and squared away" is where I get a lot of my joy from math.
But also, there are uses to math that you might be able to play with through every day, but you've never thought of those scenarios in a mathematical way.
I was walking today, and on the street there is a right angle turn. The inner portion of the turn is just a square right angle, but the outside of the turn is a radius. I started wondering to myself, if I want to be on the outside of the turn going into and exiting the turn, what would be different ways I could walk this, and what would the distance differences be.
Crossing directly across, to the inner corner and crossing directly across to the outer side again, would be 2w (for the width of the road w). Following the edge of the radius would (assuming perfectly circular), be 1/4 of a circle, so 1/42piw = 1/2 pi * w. The shortest route is a straight line, which would make a right triangle, so w^2 + w^2 = c^2, 2w^2 = c^2, sqrt(2) w = c
So crossing twice is 2w, following the edge is 1/2piw, and shortest path is sqrt(2)*w. Not super applicable, and I didn't need to do math to figure it out, but I was walking and bored, so I found joy in it. The fact that they all boil down to having w as a factor means I could figure out a nice ratio between all of them. And then I needed to mentally figure out what 1/2 pi was. 3.14/2 = 1.57... And I know that sqrt(2) is roughly 1.41 ish.
So now I know that crossing twice has a cost of 2, following the edge is 1.57, and direct line is 1.41. Following the edge is vaguely close enough to the ideal path to warrant not walking into the street to optimize the route, 1.57 / 1.41 is about ~110%. Whereas by defintion, a cost of 2 is going to be sqrt(2) times sqrt(2), so ~141% more than shortest path.
A few things to note here. First off, I'm aware that not everyone finds the same joy in doing simple mental math and thinking about problems mathematically even when there is no need to do it, but trying to think of things more minor trivial things mathematically may cause you to at least appreciate it more, which can grow into joy. And second, I wasn't doing any complicated math in my head. I just thought to myself "is it faster to cut to the inside corner and then cut back out... of course not, right?" and I was able to answer that definitively to myself. Did it matter? Was the answer probably obvious anyway? Probably, but I was able to _prove_ that. And I value facts. Finding joy in the simple things lets you build up more of a familiarity and view it more as a problem solving tool than a tedious thing to rote memorize.
A great way to build up math familiarity and see how other people find joy in mathematics would be to watch Numberphile videos on YouTube[0]. It's a bunch of mathematicians sharing things they find interesting about math. Some times are REAL hard to grasp, but some are just very interesting puzzles[1]. The puzzles don't always have clear immediate usefulness, but can often be described as "a mathematician wanted to know an answer, so they did some math to find out and prove something to themself."
Sorry, end of spiel.
tl;dr - find the joy in the simple things and use math as a tool to answer (even simple) questions to help highlight the usefulness.
0: https://www.youtube.com/channel/UCoxcjq-8xIDTYp3uz647V5A 1: https://youtu.be/ONdgXYEBihA
I didn't particularly find (at the time) calculus, multivariable calculus, physics, etc. interesting as I didn't find the applications interesting at the time. I find these subjects representative of what you traditionally learn at school.
When I entered uni I discovered my passion for discrete math, algebra (groups, rings, fields, etc.), number theory, cryptography, theory of computation, etc. as they have a lot of application in CS.
That's really what did it for me - and also I had great uni lecturers. I wish they would have taught the subjects I like in highschool - the difficulty level is about the same.
For whatever reason, many University programs use high level math classes as a filter to weed out 1st year students from that program. If university instructors had a genuine passion, and ability, for teaching high level math then they wouldn't accept that as an outcome.
The only thing I disagree with in your comment is about the instructors: they want to be employed, and they have to accept the syllabus and testing standards. It is not about passion and ability to teach (most, especially younger ones, are full of those); it is about meeting the departmental requirements.
You are misrepresenting what's happening. Other departments use beginning math classes as a way of weeding out students they feel won't succeed in their fields because they can't pass basic mathematics classes. Most math departments would absolutely love to have more students in them.
The problem is that these students aren't prepared properly by K-12 mathematics courses and math builds upon itself. If you don't have a good grasp of algebra, you just won't succeed at calculus. We're sticking people in the equivalent of Spanish 4 without having learned Spanish 1 properly.
[0] https://plus.maths.org/content/john-haigh
Consider doing something that actually needs it. You like computer programming - consider making a game engine. It might be easier to learn when you can actually see that it is useful.
Keep in mind though that math is a lot of things. People obsess over calculus but that is just one type. Math is just as much the different types of symmetry in wall paper patterns as it is finding the derrivative. Don't be afraid to try different areas. If you dont know where to start, consider picking up "A Concise Introduction to Pure Mathematics" by liebeck which introduces a bunch of different math concepts and see if any feel more interesting to you.
The tricky bit is often that you need to learn some of the math before you can see how it's useful, but if you need stronger motivation, you might try diving into a slightly math heavy programming problem and learn the math as you go
Yes, you need some practical math as well. I did engineering, there's a lot of inelegant stuff there.
But that stuff actually tends to be right next to some very interesting things.
Here are three things you can find out.
First, there's more than one kind of infinity. You can't make a map from natural numbers like 1, 2, 3 etc to real numbers like e, 0.632268, sqrt(2) etc. Look for Cantor diagonalization.
Second, a random walk like a heads vs tails comes back to zero almost certainly. It also does so in two dimensions, like walking randomly in Manhattan. In three dimensions, it does not, and so for higher dimensions. Look for Polya.
Third. There is a way for you and me to communicate secretly, despite everyone in HN being able to see our entire exchange. Look for Diffie Helmann.
These days, there's a whole industry of people doing math videos with interesting stuff.
So I think a good motive for math study is really in games and puzzles, where the questions posed aren't about win/lose or right/wrong, but about exploring the scenario further and clarifying the constraints or finding an interesting new framing. Martin Gardner wrote a long-running column and a few books in this vein which are still highly regarded decades later.
If anyone had a guaranteed way to make people enjoy math, we'd already be applying that method.
Just read ahead to figure out what you'll need to learn, and do some advance reading. Anything thag make the courses easier will tend to make them more fun.
A friend of mine taught remedial math at UW to incoming freshmen. She would write:
on the blackboard and ask a student "what is the value of x?" The student would see the x, and immediately respond with x means algebra, algebra is hard, I cannot do algebra.So she started writing:
and ask the student to fill in the blank. "Oh, it's 3!"When teaching addition, workbooks commonly use a box, eg, “[ ] + 2 = 5” — and first graders have no conceptual problem with this. Somehow, we lose people by the time we’re trying to formalize the same concept in algebra. There’s been many times I’ve written a box around letters in a problem and asked students “what’s in the box labeled x?”
Pedagogy is hard.
Numbers and letters are taught together, but not as symbols. Letters are taught with sounds and numbers are taught with counting. The notion of a symbol isn't really emphasized much.
I would explain it more like after
[ ] + 2 = 5
what happens if you need more than one box for a complicated problem? Teach the idea that saying box #3 is equivalent to assigning an arbitrary letter for whatever reason you want, but that people more familiar with math prefer letters because they stand in for words that describe what the number is for. You might want to use 'c' for the number of cats you're trying to figure out.
In a room of five animals two are dogs. How many cats?
a = 5, c = ?, d = 2
a = c + d
so... 5 = c + 2
what is c?
Light bulb goes off: "You can do that?" Yes, you can do whatever you want and it's not all about carrying the one or whatever other rote teaching they've been given. They can get creative and be engaged, and then you let them know that actually there are some conventions people like to use for what they're trying to do. They might even believe they've invented a new idea. At least they're having fun.
To me, a lot of pre-college math education could be summarized as "In this class I will show you a bunch of abstract problems, a bunch of ways to solve them, and I will test if you have learned them." Learning in these classes is often limited to memorizing a sequence of steps.
That's why I would frequently ask "You can do that?" myself when talking to those whom I considered mathematically gifted (math olympiad winners and such). I think they realized that as a problem-solving tool math could be used creatively. I saw it as a largely useless hammer that to work had to be held in a very specific way.
I remember connecting sets in, I think, Pascal to what I had learned in school and realizing that all that math was perhaps not as useless as I had thought : - )
> Go from "[ ] x 2 = 10" to writing it "box x 2 = 10--what is box?". Then "b x 2 = 10--what is b?" then "x x 2 = 10--what is x?".
From memory, we didn't switch from "x" to dot for multiplication until at the exact same time we started using symbols. If we'd done it earlier (or even right from the start) it might not have been as much of a problem.
X = X + 1
once we got it, it was a like new world!
not only is this overloading a symbol (equality) with a completely different meaning (assignment), it is also a poor choice typographically, as it represents a directional operation with a directionless symbol.
using an arrow for assignment is much better.
it's also worth pointing out that unlike most others, logic programming languages (e.g. prolog) have actual variables, not references to mutable or immutable memory cells.
for this reason, i felt C a breath of fresh air cos u could just assign using = instead of what we was doing in pascal which was the horrible := where u had to press SHIFT for the :
things like this matter.
https://upload.wikimedia.org/wikipedia/commons/9/94/APL-keyb...
People just want to know why it's x and not something else or how a letter can have value. They might even think how can 24 + 2 = 5? They just want something to grab onto and nobody is really teaching the concept of a symbol in a math class.
This was what? 5th grade?
What kind of crap teachers never taught that
It's rarely the fault of the teachers.
The problem is, in many MANY MANY schools, teachers are more like social workers that have to compensate for utter horrifics outside of school. You got a ton of children so poor they didn't have breakfast which means their first (and all too often: only) meal will be the school-provided lunch (Covid showed that - a bunch of schools were open at least for lunches). You got children that are literally homeless and living with their parents in some car on a Walmart parking lot. You got children whose parents are in and out of jail. You got children living with their siblings in way too small, pest and mold ridden "apartments". You got children whose parents don't have money to pay for basic school supplies. You got children who are dealing with mental, physical and sexual abuse. You got children where the parents are constantly on drugs or seeking for drugs. You got children with a drug dependency on their own - if they're lucky it's just tobacco or weed, if not it's opioids. You got children with parents or siblings with serious mental or physical health issues. Or you got children with their own mental and physical health issues, or if you want it worse, children with these issues but without access to any kind of treatment. You got children that are being weaponized in nasty divorces. You got children that are being weaponized by street gangs. You got children committing crimes from petty theft to dealing drugs just to survive. You got children that have to literally work (and states like FL pushing to have more working children). You got children having their own children already (either from sexual abuse, from under-education about their own bodies, or intentionally because they fell for some stupid challenge/dare). You got children dealing with bullying, you got some who actually are bullies because they have no other way of dealing with their emotions or getting lunch money. You got children with parents with about zero interest in them. You got children who worry that they'll come home and find out their parents got snatched and disappeared by ICE. You got children who worry that ICE will storm their classroom and deport them. You got children who worry they might not survive the school day because someone will shoot at them. You got children who are constantly on the move because their parents' employment/deployment requires absolute mobility. You got children who are LGBT and have to deal with ever increasing hate against them (and LGBT youth already had significantly higher suicide rates than before the GQP made it a culture war issue).
The US doesn't have any kind of system to help these children but schools and libraries, both are horribly underfunded (there's some school districts where teachers gotta take up second jobs because the government can only afford paying them for 4 days a week), and all too often teachers have to pay with their own money for students' school supplies.
And on top of dealing with these kind of nightmares, they actually have to try and teach these children something - even if the children in question aren't anywhere near a headspace where they can actually learn.
Education has a problem with scaling, especially at the elementary level. Sometimes people figure out a nice solution, but when you tell them "great, and now do this in every village" the problem becomes obvious. But there are kids in that village, too, and you want them to know reading and math and hopefully also something more.
Not if you actually provide the money. Europe gets this down decently well - although I'll admit, in rural areas in Germany we got some serious consolidation issues thanks to urban flight.
But at least our teachers are well paid government jobs and the job is decently attractive.
(Yes, this is a political opinion. No, do not blame me for that. Politics does not come wrapped up neatly with a bow tie in a box. If you want to debate the veracity of my claim, go do that instead.)
I'd do no such thing because you are completely correct - the only thing I'd add is that poverty, while being very dominant, isn't the only issue that desperately needs to be fixed.
Interestingly mathematical symbols in the past also regularly evolved. Then at some point we just stopped doing that and get stuck in a time which is arguably no longer especially appropriate. So we're left with rather inconsistent symbols, oft reused in different contexts, and optimized for written communication.
Seems like a pretty easy example to make practically, for map have a collection of things, say balls or black. Pick up each one and do a thing to them, paint them blue for example.
For filter do the same except have two different colour balls, if they are yellow they get thrown away, of they are blue they get put in a bucket.
A for loop doing exactly the same you would need to explain the topic at hand, as well as explain iterating an index etc...
Maps and filters also require understanding of higher order functions and the very idea of passing function around as a value. I would argue that implementing map/filter with a loop and then demonstrating how this pattern is generalized as .map()/.filter() functions is better and more accessible
Nobody in school ever tells you that there are glossaries on Wikipedia that tell you the meaning of the symbols. You're supposed to figure it out yourself using vibes.
The way mathematic notation is taught is inherently unstructured. You're expected to just get it.
I often see people make the mistake of trying to teach inappropriately abstract things to small children, because that's what the pros do, and we want the little kids become pros as soon as possible. Problem is, trying to skip the fundamentals is only harmful in long term.
First kids need to learn what all that stuff means, and then we can proceed to teach them the shortcuts.
What I should really do is create myself a cheat sheet of symbols to code...
https://en.m.wikipedia.org/wiki/Glossary_of_mathematical_sym...
That said, as a person who moderately enjoyed math in high school and university, this functional notation would make me hate math infinitely more. It's would look like Lisp, which, at high level, looks just as cryptic as algebra. The sheer amount of braces and mistakes that would be made when reading and writing them is nauseating.
Infix notation, for all its flaws, provides important visual aid for understanding the structure of the expression (the sum of two fractions looks very different from fraction of two sums for example). Whereas with functional notation it's like working on linear textual representation of abstract syntax tree. Trust me, nobody wants to read, write or transform one by hand
https://aplwiki.com/wiki/Comparison_with_traditional_mathema...
;-)
Also it should be sum(x+3, x, 1, 4) since you need to encode what the iterator variable is as well.
Edit: Oh wait, someone else mentions map/filter, did they mean this as a combination of range->map->sum and the latter two numbers are the range portion, like sum(map(x+3, 1..4)) ?
Edit2: And now I'm remembering sigma. I think it would have been more obvious to me if the order was flipped and your issue handled the way it is in that notation: sum(x=1, 4, x+3), though I'd still prefer the range notation: sum(x=1..4, x+3)
Granted I always found sigma a bit quirky for separating the range ends like that. Either x below and 1..4 above, or x=1..4 below/above would have been more intuitive.
But it's just a notation you learn once and then you know it.
We did these sorts of problems for a long time, with addition/multiplication/fractions, and even when we started doing actual algebra the problems were introduced the same way “let’s look at a problem we’ve solved already, and write it in a different way”.
Yeah, lots of existing math texts will forever exist with greek alphabet soup, but we don't have to rely on those as our be-all-end-all teaching tools.
To operate at a high level in mathematics I would agree that having the skill of easily abstracting complex things into compact symbols is a necessary skill, just as I would agree to the same concept applied to software engineering or really any complex engineering system; by the same token, we don't have to START on hard mode with all of our students. Math is infamously difficult for some, largely (I think) because we make it unnecessarily opaque out of some misguided sense of traditionalism.
If we want to have lots of people who are good at math we should embrace whatever pedagogy is effective.
But I've got to say, the short names are not the problem. If you rewrote F=ma as "force is equal to mass multiplied by acceleration" this wouldn't suddenly make it more accessible to swathes of the population. People who are good at maths anyway have no problem with this.
are you really saying that "let function(argument has type RealNumber) has type RealNumber be a function from a real number argument to a real number" is somehow superior to "let f(x) : R->R"
I don't know why this is a startling take. If you encode your ideas in an unfamilar symbology, of course that's going to make it more difficult for someone who isn't familiar with the space.
I'm not arguing that we should teach real analysis this way, or any other high level math class. I'm only contesting GP's comment that there is NO value to be had in using more familiar language to explain a new concept to an unfamiliar audience.
This is the entire reasoning behind "word problems" at the elementary level; they're meant to ground the abstract modeling of a math problem (193 - 3 * 12 = ?) into something more intuitive for a child to understand (If you start with 193 eggs, and you take three dozen away to bake a cake, how many are left?)
> are you really saying that "let function(argument has type RealNumber) has type RealNumber be a function from a real number argument to a real number" is somehow superior to "let f(x) : R->R"
No, I'm saying the there's tradeoffs on either side, and our educators ought to be aware of this.
> math notation is by and for professional mathematicians.
I agree, but we teach math to plenty of people who aren't professional mathematicians. I wouldn't want to do formal abstract algebra proofs in a more verbose form, I'm perfectly happy using the domain notation, but my friends from biological sciences who have to take a calculus course now have to learn both a new symbology alongside the problem domain. I've watched enough of my (clearly intelligent) biology friends slam face first into calculus and spin out. They're not dumb, they can do circles around me when it comes to chemistry, yet they Just Can't wrap their heads around calculus-style math, which leads me to wonder what the difference is between how we teach complex chemistry vs complex math. Questioning the pedagogy is a fairly logical extension of that.
It's hard for me to imagine it without any special notation. It makes me think of the general relativity in words of four characters or less that was posted recently. Sure, it might be possible, but does it really make it easier to understand? Understanding is normally built up in layers. We learn things using big words because that makes it easier than learning with small words. And we learn maths with funny symbols because it makes it easier than learning it with words (or colours or mime or other things you already know).
In fact the analogy to music notation is I think a fairly strong one. People's complaints always sound to me like asking why we don't write "C3 sixteenth note" for music instead of using dots and lines. After all, how are we meant to know what the dots mean and remember the difference between an eighth and sixteenth, or what flats/sharps do? And then the key signature can modify all of it!
The notation just isn't a barrier. Once you learn to read it, it's there because it's a clearer way to write the ideas. The hard part for people is they don't understand the ideas, and don't have the frameworks like key signatures, chord progressions, and meter to place them within. Longer words for variables won't help people understand e.g. inner and outer regular measures, or the open cover definition of compactness. That comes from a lot of work to understand what you're trying to say, the pitfalls of saying it wrong, and precisely how your slightly different way of saying it avoids those pitfalls (or selects the best set of pitfalls if you must pick some kind of degenerate behavior).
Broadly I agree; the semantic density of domain-specific language is often required to operate well in that domain. I disagree some with the "Math doesn't start on hard mode," but I think that's just bikeshedding at some level.
The endemic "I just don't understand math" that my (American) peers have espoused, to me, points to a failure in our (American, public school) instruction practices around it.
The way I was taught it and the way that worked now for my now 3 year old is just to say pirates buried a number under the X, and that we need to guess what they buried. If the concept of a number being hidden is a barrier to understanding for anyone they have seriously bad teachers.
Even if the system was better the person still has to be able to motivate themselves and put in the time.
One of my school math teacher had the same approach in another way: We were expected to use greek letters, not latin ones.
Same reasoning: It showed us kiddos that the letter was insignificant compared to the concept expressed by the letter.
So my take would be: Your friend taught the students for the first time what they were actually doing while handling equations with "a letter in it". That is no problem of algebra in itself. It just means their previous teachers sucked.
She loves it. It uses a ‘?’ for basic algebra style problems and after a few days of playing (if/when she wants to, we don’t make her play it), she was already much better and faster at those problems. It made me think that schools should be giving kids games like these.
https://www.amazon.com/Educational-Insights-Math-Electronic-...
> I have an internal KPI: if in the last three days I haven’t spent at least 30 minutes playing with my kid, there’s something seriously wrong
I think I'm interpreting this ungenerously, because my knee-jerk reaction was to wonder about who is handling the other 12+ waking hours a day.
Even with a "normal" mind. Train consistently to gain excellence!
As kids get older, they need to learn how to struggle and overcome struggles. (I would still caution against "forcing" math.) But yes, you need to start engaging hard work and determination.
Btw, the two are not mutually exclusive. Young children should be praised for struggling at things so they begin learning that skill, too.